1. Analog Filters: Doubly-Terminated LC Ladders Franco Maloberti

2. Analog Filters: Doubly-Terminated LC Ladders2 Basic Formulation The basic arrangement is The voltage or the current are no more meaningful What matter is the power delivered to the load

3. Franco MalobertiAnalog Filters: Doubly-Terminated LC Ladders3 Power Transmission Maximum transmission If the two resistances are not equal maximum power by using a transformer Max for

4. Franco MalobertiAnalog Filters: Doubly-Terminated LC Ladders4 Transmission and Reflection The transmission coefficient is defined by The reflection coefficient is For a filter in the pass-band in the stop-band

5. Franco MalobertiAnalog Filters: Doubly-Terminated LC Ladders5 Transmission and Reflection (ii) Define the input impedance at port 1 with port 2 terminated with R 2 The two-port is loss-less

6. Franco MalobertiAnalog Filters: Doubly-Terminated LC Ladders6 Transmission and Reflection (iii) The reflection coefficient becomes Realizing a given transmission coefficient becomes realizing a corresponding Z 11 (s)

7. Franco MalobertiAnalog Filters: Doubly-Terminated LC Ladders7 Ladder Filter Design: Procedure Given a certain transmission coefficient |t(j  )| 2 Estimate the reflection coefficient |  (j  )| 2 Determine  (s) Chose the upper or lower sign Realize Z 11 (s)

8. Franco MalobertiAnalog Filters: Doubly-Terminated LC Ladders8 Example den=[-1 0 0 0 0 0 1]; »b=roots(den) b = -1.0000 -0.5000 + 0.8660i -0.5000 - 0.8660i 0.5000 + 0.8660i 0.5000 - 0.8660i 1.0000 »c=poly([b(1:3)]) c = 1.0000 2.0000 2.0000 1.0000 Long division or other Implementation methods

9. Franco MalobertiAnalog Filters: Doubly-Terminated LC Ladders9 Unequal terminations Three methods Implement the problem with equal terminations Use a transformer Account for the gain reduction in t(j  )

10. Franco MalobertiAnalog Filters: Doubly-Terminated LC Ladders10 Use in Reverse Assume to exchange the role of the two ports We can flip the network The transmission coefficient in reverse is the same as the one in direct configuration (no demonstration)